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Parallel Full Matrix Algorithms

This CPS615 module covers basic full matrix parallel algorithms with a discussion of matrix multiplication, LU decomposition with latter covered for banded as well as true full case

Matrix multiplication covers the approach given in "Solving Problems on Concurrent Processors" as well as Cannon's algorithm.

We review those applications -- especially Computational electromagnetics and Chemistry -- where full matrices are commonly used

Of course sparse matrices are far more important than full matrices!

Table of Contents for Parallel Full Matrix Algorithms

001 Full Matrices
    CPS615 Computational Science for Simulation Applications
    December 4, 1995
002 Abstract of Full Matrix CPS615 Module
003 Review of Matrices seen in PDE's
004 Examples of Full Matrices in Chemistry
005 Operations used with Hamiltonian operator
006 Examples of Full Matrices in Electromagnetics
007 Computational Electromagnetics Formalism I
008 Computational Electromagnetics Formalism II
009 Comments on Computational Electromagnetics
010 Summary of Use of Full Matrices in Chemistry
011 Notes on the use of full matrices
012 Full Matrix Multiplication
013 Sub-block definition of Matrix Multiply
014 Some References
015 The First Algorithm
    (Broadcast, Multiply, and Roll)
016 The first stage -- index n=0 in sub-block sum -- of the algorithm 
    on N=16 example
017 The second stage -- n=1 in sum over subblock indices -- of the 
    algorithm on N=16 example
018 Second stage, continued
019 Look at the whole algorithm on one element
020 Cartesian Topology in MPI -- General
021 Cartesian Topology in MPI -- Matrix Multiplication
022 Cartesian Topology in MPI -- MPI_CART_SUB
023 Cartesian Topology in MPI -- MPI_CART_shift
024 Cartesian Topology in MPI -- MPI_SENDRECV_REPLACE
025 Cartesian Topology in MPI -- MPI_Cart_Create
026 Matrix Multiplication MPI Style Pseudocode
027 Matrix Multiplication Pseudocode, continued
028 Broadcast in the Full Matrix Case
029 Implementation of Naive and Log Broadcast
030 The Pipe Broadcast Operation
031 Schematic of Pipe Broadcast Operation
032 Performance Analysis of Matrix Multiplication
033 Cannon's Algorithm for Matrix Multiplication
034 Cannon's Algorithm 
035 The Set-up Stage of the Algorithm
036 The first iteration of the algorithm
037 Performance Analysis of Cannon's Algorithm
038 Full LU Decomposition
039 Some References
040 Sequential LU Algorithm
041 Sequential LU Algorithm, continued
042 Sequential Pseudocode
043 Parallel Decomposition
044 Better Parallel Decomposition
045 Parallel Pseudocode
046 Performance Analysis of the Parallel LU Decomposition
047 Banded LU Decomposition
048 Some References
049 Banded Matrix Decomposition